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Green’s Function Retrieval with Absorbing Probes in
Reverberating Cavities
Matthieu Davy, Julien de Rosny, Philippe Besnier
To cite this version:
Matthieu Davy, Julien de Rosny, Philippe Besnier.
Green’s Function Retrieval with Absorbing
Probes in Reverberating Cavities. Physical Review Letters, American Physical Society, 2016, 116
(21), pp.213902. �10.1103/PhysRevLett.116.213902�. �hal-01343529�
Matthieu Davy
Institut d’Electronique et de T´el´ecommunications de Rennes, UMR CNRS 6164, Universit´e de Rennes 1, Rennes 35042, France.∗
Julien de Rosny
ESPCI ParisTech, PSL Research University, CNRS, Institut Langevin - 1 rue Jussieu, F-75005, Paris, France
Philippe Besnier
Institut d’Electronique et de T´el´ecommunications de Rennes, UMR CNRS 6164, INSA de Rennes, Rennes 35708, France.
(Dated: March 7, 2016)
The cross-correlation of a diffuse wavefield converges toward the difference between the anti-causal and anti-causal Green’s functions between two points. This property has paved the way to passive imaging using ambient noise sources. In this letter, we investigate Green’s function retrieval in electromagnetism. Using a model based on the fluctuation dissipation theorem, we demonstrate theoretically that the cross-correlation function strongly depends on the absorption properties of the receivers. This is confirmed in measurements within a reverberation chamber. In contrast to measurements with non-invasive probes, we show that only the anti-causal Green’s function can be retrieved with a matched antenna. Finally, we interpret this result as an equivalent time-reversal experiment with an electromagnetic sink.
The Green’s function retrieval technique is now widely used for passive imaging from ambient noise. It is based on the cross-correlation of a diffuse wavefield with an array of receivers. The fluctuation-dissipation theorem (FDT) shows that the cross-correlation at two points of a field at thermal equilibrium is proportional in the frequency domain to the imaginary part of the Green’s function between them [1–5]. This property has been extended to non-thermal noise sources for which the equipartition principle is fulfilled [4–9]. In the time do-main, the anti-causal and causal Green’s function are ob-tained so that the impulse response between two receivers can be reconstructed for sufficiently broadband signals. The Green’s function can be reconstructed from thermal radiations in a cavity [4, 10] and for a diffuse field gener-ated by uncorrelgener-ated random sources evenly distributed
in a volume [7, 8, 11, 12]. In chaotic cavities [4, 13–
15] and in random media [5, 16–18], multiple scattering increases the convergence rate of the cross-correlation to-ward the Green’s function with respect to the number of noise sources. The ballistic waves and the first echoes which are usually of interest for imaging purposes can therefore be estimated within disordered media even with a small number of noise sources.
The ambient noise correlation method has paved the way for spectacular results in seismology [19, 20] (see e.g. Ref.[21] for a review). It has also been demonstrated with acoustic [4, 22, 23], elastic [15] and electromagnetic waves in microwave [10] and optical frequency ranges [24].
Studies on Green’s function retrieval have so far fo-cused on non-invasive measurements of the field, which is for instance the case of seismic stations. This means
that the wavefield is barely modified by the presence of the sensor. In this case, the cross-correlation of an equipartitioned field probed with point-like receivers is a symmetrical function in the time domain. In electro-magnetism, efficient antennas are however absorbing and scattering receivers. For radio frequency waves, antennas
are characterized by their internal impedance Z11. The
maximum power is extracted from the field when the receiving antenna is matched with the load impedance
ZL, ZL = Z11∗. In optics, the field can be probed
with quantum dipoles or nanoantennas [25]. An
ana-log impedance concept is increasingly used to model the properties of these nano-devices (absorption, resonance ...)[26–28]. Those sensors may be used to measure the cross-density of states (CDOS) which characterizes the coherence of a wavefield independently of the source dis-tribution [29]. The CDOS is proportional to the imag-inary part of the trace of the Green’s tensor between two points. Similarly to Green’s function estimation, the CDOS can be obtained from the cross-correlation of an equipartitioned field between two positions [18]. However the presence of these nano-devices may strongly modify the field and may therefore induce an important discrep-ancy from the theoretical predictions of the CDOS in-volving only the electromagnetic fields E and H.
In this letter, we investigate the cross-correlation
C(t) = s1(−t) ⊗ s2(t) of two signals s1(t) and s2(t)
ob-tained by measuring an equipartioned field at two po-sitions with absorbing receivers. We demonstrate using the FDT that probing the wavefield with antennas with different impedances gives an asymmetrical signal in the time domain. We show that in the case of a first matched
Accepted
2 antenna and a second non-invasive probe, C(t) even
van-ishes at positive times and is proportional to the
anti-causal Green’s function between them, C(t) ∼ G12(−t).
Those theoretical results are confirmed in measurements in a chaotic cavity. We take advantage of the diffuse field generated by a single source in a mode-stirred reverber-ation chamber (RC) to retrieve the impulse response be-tween two receivers. Those results illuminate the central role of antenna absorption. Furthermore since the cross-correlation can be interpreted as a time reversal (TR) process, we show that it implies that the incoming en-ergy of a time-reversed wavefront is completely absorbed by a single matched antenna.
The FDT expresses the cross-correlation at frequency
ω of two voltages U1 and U2 at thermal equilibrium in
terms of the mutual impedance matrix Z between the ports [2, 30],
C12(ω) = 2Z1,2+ Z2,1∗ kBT (ω). (1)
Here T (ω) is the temperature of the system. This rela-tion is an extension to multiport systems of the power spectral density of the Johnson noise measured by a
re-sistor R0given by 4R0kBT (ω) [31]. For a reciprocal
sys-tem, the real part of the mutual impedance matrix is
ob-tained, C12(ω) = 4kBT (ω)Re{Z2,1}. This result not only
holds for electrical systems, but also as soon as probes linearly convert a wavefield into quantities such as volt-age/current for electromagnetic waves or force/material
velocity for acoustic waves. It is of great interest for
passive imaging since the cross-correlation of voltages at thermal equilibrium measured with two antennas in their open-circuit modes is similarly given by Eq. (1). The mutual impedance of two resonant antennas in
electro-magnetism is Z2,1= iR nT2(x2)G(x1, x2)n1(x1)d3x1d3x2
where G is the electric dyadic Green’s function and
nj(xj) is the normalized distribution of the current
within the antenna j. For two dipoles of lengths l1and l2
small compared to the wavelength, l1, l2 λ, the FDT
therefore writes C12(x1, x2, ω) = 4l1l2kBT (ω) G(x1, x2, ω) − G∗(x1, x2, ω) 2i , (2) with G(x1, x2, ω) = nT2(x2)G(x1, x2, ω)n1(x1). This
equation is the classical expression of the cross-correlation function for non-invasive pointlike probes [7– 9].
For absorbing antennas, the model of a system at ther-mal equilibrium cannot be used since the part of the en-ergy that is absorbed by the antenna is not compensated by a corresponding immediate thermal emission. Eq. (1) can hence not be applied in its form. Nevertheless the correlation matrix C(ω) can be obtained from the FDT using Thevenin’s theorem [32]. The two ports system is
equivalent of two noise voltage sources that are loaded with the mutual impedance matrix Z. Fully taking into account the coupling between the ports yields [33],
C(ω) = 2kBT (ω)Q [Z + Z∗] Q†, (3)
where the matrix Q is given by, Q = ZL(Z + ZL)−1.
The load impedance ZL is a diagonal matrix whose first
and second elements are the load impedance of the two
antennas ZL1 and ZL2, respectively. For non-invasive
probes, the currents in the antennas are weak since ZL1
and ZL2 are much larger than the elements of Z. This
gives Z + ZL ∼ ZL and C12(ω) reduces to Eq. (1) as
expected.
We now investigate the interesting case of the cross-correlation between an absorbing antenna with internal
impedance Z11 and a non-invasive probe.
Straightfor-ward calculations using Eq. (3) yield,
C12(ω) = 2kbT (ω)
ZL1∗
Z∗
L1+ Z11∗
(Z1,2∗ + ΓZ1,2). (4)
Here Γ = (ZL1− Z11∗)/(ZL1+ Z11) is the usual reflection
parameter due to the impedance mismatch between the
load impedance ZL1 and the internal impedance of the
first antenna. When the first antenna is matched, Z11∗ =
ZL1 (Γ = 0), with a small reactive part (Im{Z11}
Re{Z11}), C12(ω) = kbT Z1,2∗ . Only the anti-causal
Green’s function between the antennas is hence retrieved. This phenomenon occurs because the power dissipated by
the load impedance is equal to kbT (ω).
The analogy with time reversal [34] provides an elegant way to interpret the result of the cross-correlation [16]. The cross-correlation of a diffuse field is indeed equivalent to a TR process in which the field emitted by the first
antenna at x1 is time-reversed by the noise sources and
the field is measured on a second antenna at x2. For
non-invasive receivers, the time reversed field is the
superpo-sition of the converging wave proportional to G∗(x1, x2)
followed by a diverging wave which is in opposition of
phase −G(x1, x2), ψT R(x2) ∼ G∗(x1, x2) − G(x1, x2).
Their interference suppresses the singularity at the source and the focal spot has a width given by the diffraction limit. However when the initial source antenna is ab-sorbing, a part of the incident energy is dissipated in
the load in the second step of the TR process. Eq.
(4) demonstrates that the time-reversed field ψT R(x2) ∼
G∗(x1, x2) − ΓG(x1, x2) is not symmetrical in the time
domain because of the virtual emission of an electric field
proportional to (1 − Γ)G(x1, x2) due to the current
cir-culation within the antenna.
A matched antenna (Γ = 0) fully absorbs the energy of
the incoming field and ψT R∼ G∗(x1, x2). This relation
is identical to the “acoustic sink” model that shows that it is possible to focus waves with sub-wavelength resolu-tion [35]. Such a perfect absorpresolu-tion has also been shown
Accepted
−4 −2 0 2 4 −1 −0.5 0 0.5 1 time (ns) normalized C(t)
FIG. 1. Normalized cross-correlation C(t) obtained from measurements of the field with a non-invasive electro-optical probe translated at two positions.
possible with a lossy object illuminated with a properly designed wavefront [36, 37]. These results illustrate that a matched antenna acts as a coherent passive sink in a TR process. We emphasize that a perfect absorption can be detected in measurements only with a non-invasive prob-ing of the time-reversed field. When a second absorbprob-ing
antenna is used to probe the field at x2, the incoming
wavefront can be strongly modified and is not the per-fect time-reversed field of the source. The second pulse at positive times is then retrieved.
Those results are now confirmed in measurements in
the microwave range. Measurements are carried out
within a chaotic cavity, here a reverberation chamber
(RC) of volume V = 93.3 m3. We aim to reconstruct the
impulse response between two receiving antennas at
loca-tions x1and x2from the diffuse wavefield generated by a
single source, a third antenna at x3. The equipartitioned
field is not generated from thermal radiations but from multiple scattering at the boundary of the chaotic cavity and satisfies an equipartition-like relation [38, 39]. We
measure the transmission coefficients s13(ω) and s23(ω)
with a network analyzer in the [2-4] GHz frequency band with steps of δf=100 kHz. This frequency range is well above the first mode resonance of the RC (43 MHz) so that the field is expected to be statistically isotropic, uni-form and depolarized [40]. The cross-correlation of the
two signals is given by C(ω) = s13(ω)s∗23(ω)V02(ω) where
V0(ω) = V0is the excitation voltage on port 3.
The emitting antenna is located near a corner of the RC to reduce the contribution of the ballistic waves. To enhance the contribution of late arrivals, we compensate the exponential decrease of the envelopes of the signals in the time domain. The inverse Fourier transform of
s13(ω) and s23(ω) are multiplied by exp(t/τa) , where
τa= 1.8 µs is the average damping time of the RC, and
then cross-correlated. Despite the self-averaging prop-erty of the cross-correlation in reverberating media [41], C(t) is dominated by strong spurious fluctuations
be-cause i) a single source is used; and ii) τa is small
com-pared to the Heisenberg time of the cavity, τH ∼ 8 ms,
which is estimated from the modal density per Hertz
given by Weyl’s formula. We then average the
cross-correlation over fifty positions of a stirrer made of 6
alu-minium blades with surfaces of ∼ 750λ2. Its rotation
provides statistically independent realizations of the dif-fuse field [42] so that we expect that C(t) converges to-ward the average Green’s function of the RC consisting of the direct wave between the antennas and the first echoes that are not scattered by the stirrer. In Eq. (1),
Z2,1 must therefore be replaced by its average over the
positions of the stirrer, hZ2,1i. This change is implicit in
the following analysis of the measurements.
The cross-correlation C(t) is first measured by trans-lating at two positions an electro-optical probe of length of order of a tenth of the wavelength. The probe allows a non-invasive measurement of the field. Its translation barely modifies the diffuse field within the chamber so that C(t) is equivalent to the cross-correlation recorded with two similar probes. C(t) is seen in Fig. 1 to be a symmetrical function consisting of two pulses at -1.2 ns and 1.2 ns, in agreement with Eqs. (1)-(2).
−5 −4 −3 −2 −1 0 1 2 3 4 5 −1 −0.5 0 0.5 1 time (ns) normalized C(t) (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 Γ R (b) −8 −6 −4 −2 0 2 4 6 8 −1 0 1 time (ns) normalized C(t) (c)
FIG. 2. (a) Normalized cross-correlation C(t) for a horn an-tenna and a non-invasive probe. (b) Ratio of the maximum amplitude at negative times and maximum amplitude at pos-itive times with the impedance mismatch Γ for a UWB horn antenna (star), two other horn antennas (circle and cross), a discone antenna (square) and a log-periodic dipole antenna (triangle). The straight line is R = Γ. (c) Normalized cross-correlation C(t) for the same receiving horn antennas facing each other.
Accepted
4 The cross-correlation is then performed between a horn
antenna and the probe with the same polarization. A strong asymmetry is seen in Fig. 2a despite the isotropic nature of the wavefield within the cavity. This result may seem surprising at first glance regarding previous works which mainly consider non-invasive receiver since a strong asymmetry usually reveals a non-uniform spa-tial distribution of the field [20, 43, 44] which is definitely not the case here. However this asymmetry is fully pre-dicted by Eq. (4). It shows that the amplitude of C(t) at positive times relative to its amplitude at negative times, R = max[|C(t > 0)|]/max[|C(t < 0)|], is expected to be
equal to Γ. Γ can be evaluated from the S11 parameter
(Γ = |S11|). From Fig. 2a, we find R = 0.35 which is in
good agreement with the average over the bandwidth of Γ = 0.38. R and Γ are then measured for other antennas (a horn antenna, a discone antenna and a log-periodic dipole antenna) in various frequency ranges and are seen to be close in Fig. 2b. We find ratios R typically slightly smaller than the theoretical predictions because Eq. (4) does not include losses which are ∼10% for those anten-nas.
We also compute the cross-correlation for two similar horn antennas facing each other in Fig. 2c. The sym-metry is obtained as for non-invasive probes but is now related to the use of similar scattering and absorbing an-tennas. More generally, it can be shown using a com-puter algebra system that the ratio of the amplitudes at positive and negative times is given by the ratio of the
impedance mismatches of the two antennas, Γ1/Γ2. Note
that retrieving the Green’s function with those aperture antennas was not trivial regarding previous studies
con-time (ns) distance (m) (a) −8 −6 −4 −2 0 2 4 6 8 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 −1 −0.5 0 0.5 1 time (ns) normalized dC(t)/dt (b)
FIG. 3. (a) Colormap representation of C(t) when the dis-tance between the horn antenna and the probe increases from 0.55m to 1m. (b) dC(t)/dt (blue) taken at negative times is compared to the impulse response (red).
−8 −6 −4 −2 0 2 4 6 8 −1 −0.5 0 0.5 1 time (ns) time−reversed signal 50 Ω OC SC
FIG. 4. The time-reversed signal is plotted for three load impedances: a 50 Ω load (blue), an open-circuit load (red) and a short-circuit load (black).
sidering non-invasive receivers. Noise sources that con-tribute to the cross-correlation have indeed been shown to be in the stationary phase region [20, 43, 44] that is located in the alignment of the receivers. However, those contributions are weak for horn antennas because of their directivity patterns. The coupling between antennas is the key to interpret C(t).
We show in Fig. 3 that we accurately retrieve the
impulse response averaged over the positions of the stir-rer between two receivers, here a horn antenna and the probe. The delay time of the two ballistic pulses is seen in Fig. 3a to correspond to the travel time between the two receivers. The normalized derivative of the cross-correlation dC(t)/dt at negative times is seen in Fig. 3b to be in very good agreement with the impulse response over more than 25 ns. Not only the direct impulse re-sponse but also the first echoes in the cavity that are of interest for imaging applications are retrieved as seen around 18 ns.
To further illustrate the influence of absorption in the load, we first measure the impulse response between the first horn antenna and the third antenna. We then mod-ify the load impedance at the connector of the source an-tenna in the second step of a TR experiment. The time-reversed field is measured with the non-invasive probe. We successively add a 50Ω load which is almost matched to the antenna internal impedance, an open-circuit (OC) load and a short-circuit (SC) load. The causal part of the time-reversed signal is seen in Fig. 4 to be strongly enhanced for the OC and SC loads (|Γ| ∼ 1) in compari-son to the 50Ω load (Γ ∼ 0). The signals corresponding to the scattering from the OC and SC loads are in phase opposition [45]. This confirms that the decrease of the signal at positive times is due to absorption by the load. In conclusion, we have demonstrated Green’s function retrieval in electromagnetism within a chaotic cavity with
a single source. By modifying the modes of the
cav-ity with a stirrer, the cross-correlation of the wavefield converges toward the average impulse response between the receivers. The theoretical and experimental results
Accepted
provide a new insight into passive imaging experiments using two absorbing receivers. They illuminate the rela-tions between cross-correlation of random wavefields and scattering, absorption and impedance properties of the receivers. Our approach also provides a new framework for applications such as antenna directivity patterns mea-surements. In particular, it makes it possible to measure the coupling between two antennas that can be used in their receiving or transmitting modes only.
∗
matthieu.davy@univ-rennes1.fr
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